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Register a new userSurface area deviation between smooth convex bodies and polytopes
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The deviation of a general convex body with twice differentiable boundary and an arbitrarily positioned polytope with a given number of vertices is studied. The paper considers the case where the deviation is measured in terms of the surface areas of the involved sets, more precisely, by what is called the surface area deviation. The proof uses arguments and constructions from probability, convex and integral geometry. The bound is closely related to $p$-affine surface areas.
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Let $K$ be a convex body in $mathbb{R}^n$ and $f : partial K rightarrow mathbb{R}_+$ a continuous, strictly positive function with $intlimits_{partial K} f(x) d mu_{partial K}(x) = 1$. We give an upper bound for the approximation of $K$ in the symmetric difference metric by an arbitrarily positioned polytope $P_f$ in $mathbb{R}^n$ having a fixed number of vertices. This generalizes a result by Ludwig, Schutt and Werner $[36]$. The polytope $P_f$ is obtained by a random construction via a probability measure with density $f$. In our result, the dependence on the number of vertices is optimal. With the optimal density $f$, the dependence on $K$ in our result is also optimal.
We prove several estimates for the volume, mean width, and the value of the Wills functional of sections of convex bodies in Johns position, as well as for their polar bodies. These estimates extend some well-known results for convex bodies in Johns position to the case of lower-dimensional sections, which had mainly been studied for the cube and the regular simplex. Some estimates for centrally symmetric convex bodies in minimal surface area position are also obtained.
Let $K subset R^d$ be a smooth convex set and let $P_la$ be a Poisson point process on $R^d$ of intensity $la$. The convex hull of $P_la cap K$ is a random convex polytope $K_la$. As $la to infty$, we show that the variance of the number of $k$-dimensional faces of $K_la$, when properly scaled, converges to a scalar multiple of the affine surface area of $K$. Similar asymptotics hold for the variance of the number of $k$-dimensional faces for the convex hull of a binomial process in $K$.
Central limit theorems for the log-volume of a class of random convex bodies in $mathbb{R}^n$ are obtained in the high-dimensional regime, that is, as $ntoinfty$. In particular, the case of random simplices pinned at the origin and simplices where all vertices are generated at random is investigated. The coordinates of the generating vectors are assumed to be independent and identically distributed with subexponential tails. In addition, asymptotic normality is established also for random convex bodies (including random simplices pinned at the origin) when the spanning vectors are distributed according to a radially symmetric probability measure on the $n$-dimensional $ell_p$-ball. In particular, this includes the cone and the uniform probability measure.
We define a set inner product to be a function on pairs of convex bodies which is symmetric, Minkowski linear in each dimension, positive definite, and satisfies the natural analogue of the Cauchy-Schwartz inequality (which is not implied by the other conditions). We show that any set inner product can be embedded into an inner product space on the associated support functions, thereby extending fundamental results of Hormander and Radstrom. The set inner product provides a geometry on the space of convex bodies. We explore some of the properties of that geometry, and discuss an application of these ideas to the reconstruction of ancestral ecological niches in evolutionary biology.
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